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This paper presents a novel dynamic bandwidth allocation (DBA) scheme, called SLICT, for service level agreeable EPON. To guarantee the upper bound of maximum polling interval, we introduce sliding cycle time constraint. A new concept of shared time and remnant time is used to calculate extended bandwidth allocation. For service level agreeable EPON, we propose a design method to support multiple services such as fixed, guaranteed and extended bandwidth service. We show that simple remnant time management based on greedy contention guarantees steady state fairness. Simulation results show that proposed scheme outperforms existing schemes in aspects of throughput, delay, loss, and average queue size under self similar traffic.